Dealing with Life Course Data in Demography: Statistical and Data ...

Dealing with Life Course Data in Demography: Statistical and Data Mining Approaches Gilbert Ritschard1 and Michel Oris2 1

2

Dept of Econometrics, University of Geneva, CH-1211 Geneva 4, Switzerland [email protected] Dept of Economic History, University of Geneva, CH-1211 Geneva 4, Switzerland [email protected]

Abstract. This paper has essentially a methodological purpose. In a first section, we shortly present demography and historical demography, the intimacy between those two disciplines and their common intellectual history, the crisis they experimented in the 1980s, and how the life course paradigm and methods have been implemented to face up the challenge of shifting “from structure to process, from macro to micro, from analysis to synthesis, from certainty to uncertainty” (Willekens, 1999, pages 2629). This retrospective look also shows impressive progresses to promote a real interdisciplinarity in population studies, family demography being probably the best example. However, we also note that the success of multivariate causal analyses has been so rapid that some pitfalls are not always avoided. In Section 2, we focus on the study of transitions. First, readers mind is refreshed about regression models, then we discuss and illustrate the problem of population heterogeneity, how it could affect results interpretation, and the interest of robust estimates and the notion of shared frailty to deal with. We also present a less popular method than event history models, however well suited for studying states observed at periodic time, the Markovian models. In Sections 3, we face the gap that we observe between standard demographic analysis and causal research, i.e. the deficit of knowledge on trajectories. The developing field of data mining provides useful tools to fill such a gap and we would like to promote their use.

1

Life course approach in demography and historical demography

Probably no discipline is more dependent of an unique model than demography. This model, if not a law, is the one of the demographic transition, elaborated by Landry and Notestein in the 1930s and 1940s. It offered a comprehensive reading of past, present and future of world population, including the disparity between developed and developing countries. Such a frame outdated both geographical and chronological boarders. Two examples are especially famous. Princeton demographers studied the fertility decline in nineteenth century Europe to find solutions that could be applied to the so-called “third world” (Coale and Watkins,

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1986). Louis Henry, a French demographer, was at the origin of historical demography tremendous development in the 1960s/1970s when he invented the family reconstitution methods to observe a “natural” - pre-transitional - fertility among the eighteenth century European rural population (Henry, 1956). Since fifty years it is quite common for demographers to work on the past and for historical demographers to work on the present, if not the future. In this paper, we stay close to this tradition, taking our illustrations indifferently in demography or historical demography. Those two disciplines share not only a foundling model but also, to a large extent, a similar intellectual history. Telling such history is of course not our purpose, but a rapid summary is important to see when and why the life course paradigm and longitudinal methods emerged. Dealing with structures and flows, demography has been a science of reconstruction and description of patterns and behaviors, through a well-established quantitative methodology, and the conviction that higher the number of observations, more accurate - and possibly useful - were the results (Pressat’s manuals remain classical for generations). Demography was a science of the masses, growing or stagnating, young or old, not of the individuals. In the same time, the engagement of generations of scholars was largely motivated by the central character of population issues and the location of demography at a crossroad between economy, sociology, epidemiological studies, territorial analysis, political sciences and, more recently, cultural and gender approaches. However, research and collaborations were in reality highly segmented, with a clear tendency to specialization on a geographical and/or thematic basis (typically, mortality, fertility, marriage and family formation or dissolution, migrations, structures, prospective). Demography, and to a lesser extent historical demography, hesitated between the temptation of autonomy, often associated with a closing on its quantitative core, and its disappearance within the social sciences, with in-between the development studies or the “population sciences”. A real intellectual crisis resulted from such hesitation, as well as from the frustration against segmentation, and also from a growing conscience that description, especially some quantification with a pretension of objectivity, hid and diffused ideological visions about what could be a “good” or “optimal” population (V´eron, 1993). Among the many reactions, revisions and re-examinations, new approaches and new methods progressively emerged. Something that retrospectively could seem very strange but is a perfect illustration of our assertions in the preceding lines, is the discovery in the 1980s of an almost complete absence of dialogue between demography and family sociology. While family is the place where most of the demographic behaviors took place and, to some extent, are decided, “few textbooks on population contain a chapter devoted to the demography of the family. Where such chapter does exist, it is generally shorter and more superficial than those that deal with fertility, mortality, nuptiality, and migration, or with the dynamics of age structure” (H¨ohn, 1992, p. 3). In 1982, the International Union for the Scientific Study of Population created an ad hoc committee to develop its study, but still in 1992 the animators of this group

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saw family demography as “a recent and relatively underdeveloped branch of population studies” (Berquo and Xenos, 1992, p. 8). Its development has been extraordinary in the last years and is part of a shift from macro to micro, from an emphasis on macro-economic changes as the essential determinant of demographic changes to a multi-causal - multivariate approach of behaviors, from average results to the study of distributions. In a quantitative discipline, major evolutions necessarily imply to take up technical challenges. “The traditional demographic analysis of such events as births, marriages, divorces, deaths, and migration, has the advantage that numbers of these events can be related to individuals in the same age group and can, therefore, be measured more easily and included in models. The inclusion of other family members in such analyses causes difficulties because they will generally differ in age and sex, and complications are also introduced because they do not generally live together continuously” (H¨ohn, 1992, p. 3). Although several attempts have been done to construct a “household demography” (Van Imhoff et al., 1995), the life course paradigm clearly imposed itself. Offering both concepts and statistical methods in an explicitly interdisciplinary perspective, it deeply renew the discipline, representing a shift toward micro analysis of individual data and causal research (Dykstra and van Wissen, 1999). A first substantial gain has been the study of multiple events, marriage and first birth, or moving and starting a new job for instance, a kind of investigation that also raise the issue of event sequencing and interactions that is typically treated with event history analysis. If people have several careers that they must make compatible, their life transitions also reflect socio-economic constraints, cultural norms (about the “proper” age, sex or behavior), as well as compromises between several individual aspirations within or beyond the domestic unit. Through researches in this huge area, family demography made for sure tremendous progress during the last 20 years. Our strong feeling is that the shift has been so sudden that globally the complexity of causalities is too often under-estimated (see especially Courgeau and Leli`evre, 1993; Blossfeld and Rowher, 2002; Bocquier, 1996; Alter, 1998), as well as several technical traps. The problem is essentially that when studying a population of individuals observed along the time, since each life, product of complex and multiple interactions, is as a matter of fact unique, interpreting and generalizing from samples requires several cautions. In the next section, we remind the main event history regression models and discuss the question of heterogeneity. We cannot consider that the elaboration of indicators at an individual levels about household, family and community contexts is enough to deal with the more and more raised issue of “linked” or “interdependent” lives (Hagestad, 2003). We show the interest of robust estimates and shared frailty in that perspective. In the same section, we also present the Markovian models that are particularly useful for the study of transitions within a set of states (social status, for example) periodically observed. In the interdisciplinary perspective that is the one of life course, we consider important to go beyond the simple transitions typically studied in demography (from single to married, from a first

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to a possible second child, from life to death, and so on) and to investigate how, from a starting position, a destination is selected among several possible. While family dynamics and life courses are more and more open, such investigations are essential to deal with the characterization of transitions as “normal” or “nonnormal” without falling again in the trap of ideological reading (see, for example, Oris and Poulain, 2003). Indeed, we assess more globally that between aggregate descriptions and causal analysis there is an obvious deficit of research on trajectories. Regression models indicate the probability that a factor, measured by an indicator, affect a risk, but such results tell us nothing about the calendar and no more about the alternatives to this risk in life courses. It is essential to look carefully at transitions in trajectories to properly target a causal analysis, and this step is clearly too often superficial, if not absent. Several methods, recently developed or recently made available in statistical packages, offer opportunities to fill this gap. In Section 3, we introduce the data mining approach, especially mining event sequential association rules and the use of induction trees.

2

Statistical modeling of life events

Life courses data are longitudinal in their essence. Here, we focus on events, an event being the change of state of some discrete variable, e.g. the marital status, the number of children, the job, the place of residence. Such data are collected in mainly two ways: as a collection of time stamped events or as state sequences. In the former case, each individual is described by a collection of time stamped events, i.e. the realization of each event of interest, e.g. being married, birth of a child, end of job, moving, is mentioned together with the time at which it occurred. In the second case, the life events of each individual are represented by the sequence of states of the variables of interest. Panel data are special case of state sequences where the states are observed at periodic time. The first kind of data is typically analysed with event history regression methods, while methods for state sequence analysis like Markov transition models are best suited for the latter. We briefly discuss hereafter the scope and limits of these approaches. 2.1

Event history regression models

When we have time stamped events, the question of interest is the duration of the spell between two successive events, or somewhat equivalently the hazard rate h(t) for the next event to occur precisely after a duration t, i.e. the conditional probability for the event to occur at t knowing that it did not occur before t. Longitudinal regression models focus on this aspect. They express either the duration or the hazard rate as a function of covariates. There are continuous time models and discrete time forms. With continuous time, the main formulations (see Blossfeld et al., 1989; Courgeau and Leli`evre, 1993) are as a duration model, T (x1 , . . . , xp ) = T0 exp(β1 x1 + · · · + βp xp ) ,

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or as a proportional hazard model h(t, x1 , . . . , xp ) = h0 (t) exp(β1 x1 + · · · + βp xp ) . The former, also known as the accelerated failure time model, assumes usually an exponential, Weibull, log-normal, log-logistic or gamma distribution for T . The proportional hazard model is compatible with for instance, exponential, Weibull and Gompertz duration distributions. It includes also the perhaps most widely used Cox (1972) semi-parametric model that requires no assumptions on the form of the duration distribution. Most statistical packages (SAS, S-Plus, Stata, R, TDA, ...) provide procedures for estimating such models. SPSS, however, offers only support for the Cox model. Discrete time models (see Allison, 1982; Yamaguchi, 1991) include the proportional hazard odds ratio model, also due to Cox (1972), ht (0) ht (x1 , . . . , xp ) = exp(β1 x1 + · · · + βp xp ) 1 − ht (x1 , . . . , xp ) 1 − ht (0) and the log-rate model (Holford, 1980) ht (x1 , . . . , xp ) = exp(β1 x1 + · · · + βp xp ) In the latter case, the xi ’s are usually dummies coding categorical variables, their interactions and, possibly, interactions with t. For the estimation of the proportional hazard odds ratio model, some assumptions are usually required upon the baseline hazard odd. Letting β0t be the baseline log-hazard ln[ht (0)/(1 − ht (0))], the most current assumptions are β0t = β0 (constant), β0t = β0 t (linear with t, Gompertz), β0t = β0 ln t (linear with ln t, Weibull). With these assumptions, a proportional hazard ratio model can, if we organize the data in a person-period form, simply be estimated as a logistic regression. Hence, it can be estimated by any software that proposes logistic regression. Likewise, a log-rate model can be estimated with any log-linear model procedure that allows for weighted cell frequencies. Indeed, the log-rate model is a log-linear model of the weighted number of events occurring in a time interval, the weight being the inverse of the population at risk in this interval. A common issue with the time to event models is the handling of censored data. Censored data occur when the observed start (left) and/or end (right) time of a spell are not its actual start and end time. For instance, if we observe job duration, some jobs may not be terminated at the time of the survey and are hence right censored. Though no event is recorded at the end of the right censored spells, these cases are taken into account by entering the population at risk for job length lower or equal to the observed duration. Another issue is the handling of time varying covariates. The solution is quite straightforward in the discrete time setting that works on person-time data. For the continuous case, there are two major solutions: an ad-hoc extension of the Cox model that allows for discrete time varying covariate and the episode-splitting approach. (See for instance Blossfeld and Rowher, 2002, for details.) For more advanced developments of the Cox model see Therneau and Grambsch (2000).

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This event history modeling, especially the Cox proportional hazard and the Cox discrete time proportional hazard odds ratio models, has become popular among demographers. Together with other social science scientists, historical demographers have to face issues like competing events (multiple destinations), repeatable events and interacting events. The first two can easily be handled with a software like TDA (Rohwer and P¨otter, 2002) that supports episodes defined by 4 parameters, namely the origin state, the start time, the destination state and the end time. The interaction between events, marriage and first child for instance needs a simultaneous equation approach that has been investigated for instance by Lillard (1993). Shared heterogeneity and multi-level modeling. A further issue of importance, shared heterogeneity, has to do with the sampling nature of the data. These are often clustered, i.e. the individual data come from a selection of groups, parishes or families for example. In such cases, members of a same group share a same contextual framework and it is then of primary importance to distinguish effects that hold at the group level from those that work at the individual level.

9 y = 15.6 - 0.8 x

8 y = 12.5 - 0.8 x 7

Children

6 5 4 3

y = 3.2 + 0.2 x

2 y = 6.2 - 0.8 x

1 0 1

3

5

7

9

11

13

15

Education

Fig. 1. Multi-level: A simple example with 3 clusters

To explain this aspect, let us consider the case of a simple linear regression of the number of children on the education level in the presence of three clusters like those depicted in Figure 1 where the clusters are, let us say, three villages. A simple regression on the whole data set leads to a line with a positive slope, indicating that the number of children increases with education. This effect clearly holds at the aggregated village level, i.e. the higher the average education level in a village, the higher the average number of children. This aggregated effect results despite the regression is fitted on individual data. A separate regression

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on each cluster exhibits a negative slope in each of the three villages, indicating a negative effect of education on the number of children at the individual level. Indeed similar misleading results may appear when event history regressions are fitted on clustered data. What are the solutions? Table 1. Alternative linear models in presence of G clusters g Model m1 yig = a + bxig + uig

Stdev of u σ

#

coef.

2+1

average model

G(2 + 1)

independent

m2 yig = ag + bg xig + uig

σ1 . . . σG

m3 yig = ag + bg xig + uig

σ

2G + 1

seemingly indep.

m4 yig = ag + bxig + uig

σ

G+2

dummies

σa , σb , σ

2+3

random effects

σa , σ

2+2

shared frailty

m5 yig = (a + uag ) + (b + ubg )xig + uig m6 yig = (a + uag ) + bxig + uig

Table 1 summarizes alternative formulations that can be adopted when we are in presence of G groups. For the sake of simplicity we consider simple regression models, generalization to more complex models like event history models being straightforward. Model m1 will capture effects at the group level. In models m2 to m4, differences between groups are introduced by means of additional parameters, an approach that is suitable as long as G is not too large. Model m2 fits separate models on each clusters, while in m3, the regressions are only seemingly independent, since the variance of the error term uig is supposed to be the same in each group g. Model m4 corresponds to the well known case where, for each group, a specific effect is introduced as a dummy variable. For a large number of groups, random effects models m5 and m6 are best suited. In these models, the regression coefficients are allowed to vary randomly from one group to another. In the shared frailty formulation, only the constant is random, while the other coefficients remain the same for all groups. The main advantage of these random effect formulations is that their number of parameters is independent of the number of clusters. Random effect models m5 and m6 may, therefore, have a much lower number of parameters than models m2 to m4 when G is large. Even if we are interested in the aggregated effect, estimating them with individual data, as with model m1 for example, requires some caution. Indeed, the standard errors of the aggregated effects are derived from individual residuals which may either over- or underestimate the between group discrepancy. For instance, in our example of Figure 1, leaving out in turn each of the three groups leads to great variations in the slope that would be underestimated by the classical standard error. This aspect has been investigated among others by Kish and Frankel (1974) and, for the Cox model for instance by Lin and Wei (1989). In such settings, it is good practice to use robust estimates of the variance of the

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regression coefficients. Such robust estimates are usually obtained as grouped jackknife estimates, i.e. by measuring the discrepancy of estimates obtained by leaving out successively each of the G clusters, and can be expressed as sandwich estimates (see for instance Therneau and Grambsch, 2000, pages 170–173). Facilities for dealing with clusters are offered by several softwares, Stata, S-Plus and R for instance. All the three mentioned softwares propose options to get robust standard errors. They permit also the introduction of a shared frailty in parametric hazard rate and Cox models. Complete random effects is only available with discrete models that can be fitted with logistic regression procedures. Indeed, logistic models are special cases of Generalized Linear Models (GLM). Hence, multilevel logistic regression is available whenever multilevel GLM is implemented. Barber et al. (2000), show how to estimate a model with several random effects with the HLM (Bryk et al., 1996) and MLN (Goldstein et al., 1998) softwares. Illustration. To illustrate the scope of robust standard errors and shared frailty we consider a data set of 5351 migrants collected from the 19th century population registers of the Belgian commune of Sart (see Alter and Oris, 2000; Alter et al., 2001, for a detailed description). This data set provides, among others, information about the emigration date, the destination and the date of return after emigration. Table 2 shows results of the fit of a continuous time Cox model. The hazard modeled is that of return after a time between 0 and 5 years, no return or return after 5 years being censored. We fitted a basic model, i.e. without the cluster or frailty options, the same model but requesting robust standard errors for the coefficients, and the model with a gamma γ(1/θ, 1/θ) distributed frailty term νg shared by members of a same family g. Formally, the latter model is h(t|x1 , . . . , xp ) = νg h0 (t) exp(β1 x1 + · · · + βp xp ) , where νg is the shared frailty term with E(νg ) = 1 and Var(νg ) = θ. 3 The hazard ratios reported are just the exponential of the coefficients. They indicate the hazard ratio for two profiles that differ by one unit of the corresponding variable. For the frailty model, this interpretation holds assuming the two profiles have a same frailty. For instance, according to the basic model, the chances to return for a single are about one a half times the chances to return for a non single. Likewise, the probability to return is for a man about 3/4 of that for a woman. The coefficients are indeed the same for the basic and robust standard errors models. The significance of the coefficients differs however, as can be seen from the p-values. To be born in the Ardennes is significant at the 5% level when we do not care about the cluster effect, while it is clearly not when we control for it. This indicates that the seemingly significant birth place effect does not work 3

The estimations were obtained with S-plus 6.2. There seems to be a bug in Stata 8 that was not able to converge within 24 hours for the frailty model while S-Plus provided the results within 2 minutes.

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Table 2. Cox model for Return within 5 years after emigration, Sart 1812-1900, n = 5351

Economic ratio Man Single Born in Ardennes Age when Leaving To Ardennes To rural To urban/indust. To other Head or spouse of Child of head Other parenthood No parenthood Standard deviation

coefficient hazard ratio basic frailty basic frailty 1.02 0.30 2.76 1.35 -0.28 -0.18 0.76 0.83 0.40 0.52 1.49 1.68 0.25 0.17 1.29 1.18 0.01 0.00 1.01 1.00 destination reference category -0.32 -0.60 0.73 0.55 -0.07 -0.23 0.93 0.79 -1.25 -1.25 0.29 0.29 parenthood reference category 0.02 -0.25 1.02 0.78 0.12 -0.27 1.13 0.76 -0.50 -0.54 0.61 0.58 √ θ of family effect 1.75

basic 0.2 0.1 1.2 4.1 12.0

p-value in % robust frailty 3.8 45.0 0.2 5.6 1.2 0.3 15.0 28.0 17.0 62.0

5.7 50.0 0.0

14.0 68.0 0.0

0.2 6.8 0.0

89.0 54.0 6.7

90.0 56.0 7.3

19.0 26.0 9.0 0.0

at the family aggregated level. Likewise, we may notice that, though the effect of the economic ratio is significant among families, its significance is not as clear as we would expect from the basic model. Let us now look at the results with a family shared frailty. First, we may notice the highly significant variance of the random term, which clearly indicates a between families discrepancy. Two variables that looked significant become non significant, namely the gender (man) and the economic ratio. This is not surprising for the latter, which is a typical family contextual factor shared by members of a same family. Gender, on the other hand, is clearly an individual characteristic. Its lack of significance in the frailty model seems to indicate that the effect is not systematic within the families. Its overall significance follows probably from differences among male and female singles. A reverse phenomenon is observed for the rural destination effect that becomes significantly different from the reference Ardennes in the frailty model. 2.2

Markov transition models

In the presence of state sequences in panel data form, a natural question is what are the transition probabilities from the states at time t − 1 to the possible states at time t, and how are these probabilities affected by individual histories or contextual characteristics. Homogenous Markov models assume that these probabilities are independent of the time t. In first order models, the transitions are supposed to depend only upon the state at t − 1, which means that the first lag summarizes the whole history of states at t − 1 and before. Models of higher order k consider that the transitions depend on k lags, i.e. on the states at t − k, . . . , t − 1. Thus, basic Markov models state that the transition probabilities

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remain constant over time and depend on a limited, usually small, set of previous states. Markov models of order k generate, when we are in presence of s states, sk transition distributions, i.e. a huge number of probabilities. They may be approximated by Mixture Transition Distribution (MTD) models (Raftery and Tavar´e, 1994; Berchtold, 2001; Berchtold and Raftery, 2002) that involve a much lower number of parameters, which renders the models easier to interpret. Other extensions of the Markov model include the Hidden Markov Model (HMM) (see Rabiner, 1989; MacDonald and Zucchini, 1997) in which the successive states of the observed variable are only indirectly linked through an unobserved Markov chain and the Double Chain Markov Model (DCMM) (Paliwal, 1993; Berchtold, 1999, 2002) which states that the observed states are outcomes of a Markov process randomly selected by a Hidden process. The use of Hidden processes is a way to relax the usually strong homogeneity assumption. For example, when studying social mobility with data covering a whole century, it is hardly defendable to assume that the same process works during the whole period. Despite their interest, there has been only a limited use of Markov models, especially of non-homogenous ones, by historians and demographers. The main reason is that standard statistical packages offer only limited facilities to fit such models. The available tools require a heavy coding task that discourages most potential users. We can expect, however, that Markov modeling will become much more popular with the recent release of March 2 (Berchtold and Berchtold, 2004). This software offers a friendly way to estimate Markov models without writing down any line of code. Illustration. To illustrate the nature of knowledge we can expect from such an analysis, we consider here the Blossfeld and Rowher (2002) sample of 600 job episodes extracted from the German Life History Study. The episodes have been classified into 3 job length categories: (1) ≤ 3 years, (2) > 3 and ≤ 10 years, (3) more than 10 years, and the data reorganized into 162 individual sequences of 2 to 9 job episodes, dropping the cases with a single episode. The question considered is how does the present episode length depend upon the those of the preceding jobs. Notice that the job lengths sequences considered here are not panel data, which demonstrates that Markovian model are not restricted to panel data. In this setting, the subscript t refers simply to the position in the sequence rather to a specific time period. The first order and second order homogenous transition matrices are given in Table 3. The same table gives also the distribution of the independence model in which the transition probabilities stay the same whatever the previous job length. The first order matrix exhibits some differences in the transition probabilities after a short (1), medium (3) or long (3) job. After a first job, the probability to start a short job after a short one is significantly higher than to start a medium or long job, while this is not the case after a medium or long job. The second order matrix does not provide evidence on the impact of the lag 2 job length. The

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Table 3. First and second order homogenous Markov matrices job t−1 1 2 3 Indep

length at t 1 2 3 .57 .30 .13 .43 .42 .15 .20 .53 .27 .50

.35

.15

half conf. interval .10 .13 .29 .07

t−2 1 2 3 1 2 3 1 2 3

job length at t t−1 1 2 1 .55 .30 1 .60 .30 1 1 0 2 .37 .45 2 .50 .41 2 .45 .33 3 .33 .17 3 0 .87 3 1 0

3 .15 .10 0 .18 .09 .22 .50 .13 0

half conf. interval .11 .20 .65 .18 .20 .38 .46 .40 1

main differences concern the transition probabilities after long jobs (3), which are mostly not statistically significant due to the low number of cases concerned. This was confirmed by fitting a MTD model for which we obtained a weight of 1 for the first lag and, hence, 0 for the second lag. Table 4. Two state hidden Markov model Hidden state at t t−1 1 2 1 .78 .22 2 .53 .47 initial

.56

.44

half conf. interval .12 .19

Hidden state 1 2

job length 1 2 3 .75 .23 .02 .05 .58 .37

half conf. interval .12 .18

.11

For relaxing the homogeneity assumption, we consider a HMM model with a two hidden state process. Fitting this model we get the distribution of the initial state of the hidden variable, the transition matrix of the hidden process and the distributions of the transition to the job length categories associated to each of the two hidden states. These results are given in Table 4. In addition we get estimates (not shown here) of the most likely sequence of hidden states associated to each observed sequence. Looking at the cross tabulation of these estimated hidden states with the observed job length hidden 1 2

observed 1 2 3 118 19 0 0 65 35

we see that the first hidden state is mainly associated to short jobs and the second hidden state to medium and long jobs. This may suggest to consider only two types of jobs: equal or less than 3 years, and more than 3 years.

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G. Ritschard and M. Oris Table 5. Global model goodness-of-fit statistics

Tree Indep Hom. Order 1 Hom. Order 2 HMM 2 States

p 2 6 13 7

-2LogLik 472.8 462.6 460.6 468.6

number of sequences: 107,

Chi2 0 10.2 12.2 4.2

df 0 4 11 5

sig .04 .35 .52

BIC 483.7 495.4 531.7 506.9

AIC 476.8 474.6 486.6 482.6

pseudo R2 0 .022 .026 .009

usable n = 237

Table 5 summarizes goodness-of-fit statistics for our fitted models and, for the sake of comparison of the independence model. The shown statistics are the number of independent parameters p, the deviance measured as minus twice the log-likelihood (-2LogLik), the likelihood ratio Chi-Square statistics that measures the improvement in -2LogLik together with its associated degree of freedom and its significance level, the pseudo R2 that gives the relative improvement in -2LogLik and the AIC and BIC information criteria. These figures show that the fitted models do not make much better than the independence model. We get the smallest -2LogLik value for the 2nd order homogenous model, but at the cost of 11 additional independent parameters. The 1st order homogenous model is the only one that significantly improves the -2LogLik of the independence model. It is also slightly better in terms of the AIC. No model outperforms the independence model in terms of the BIC however. These relatively bad results are largely attributable here to the insufficient number of data considered. This stresses a limitation of this Markov modeling approach, namely the complexity of the models in terms of number of estimated parameters that requires a very large number of data.

3

Mining longitudinal life course data

Despite the last decade great boost in the use of data mining tools for the knowledge discovery from data (KDD) in fields ranging from genetics to finance, from marketing to medical diagnosing, from text analysis to image or speech recognition, such approaches have received only little attention for extracting interesting knowledge from longitudinal data describing life courses. An important exception is Blockeel et al. (2001) who showed how mining frequent itemsets may be used to detect temporal changes in event sequences frequency from the Austrian FFS data. In Billari et al. (2000), three of the same authors also experienced an induction tree approach for exploring differences in Austrian and Italian life event sequences. We initiated ourself (Oris et al., 2003) social mobility analysis with induction trees. Data mining is mainly concerned with the characterization of interesting pattern, either per se (unsupervised learning) or for a classification or prediction purpose (supervised learning). Unlike the statistical modeling approach, it makes no assumptions about an underlying process generating the data and proceeds

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mainly heuristically. In this section, we shortly describe the mining of sequential rules and the induction tree approach, focusing on the nature of knowledge we may expect from such tools. For a more general introduction to data mining, see for instance Hand et al. (2001) or Han and Kamber (2001). These books cover many more methods. The two discussed here are however, in our mind, the two more promising ones for longitudinal data. 3.1

Mining event sequential association rules

Each life course can be seen as a sequence of life events: birth, important disease, recovering from disease, starting school, ending school, first job, first union, leaving home, first child, death of father, marriage, ... Mining sequential association rules aims at determining the most typical sequences or subsequences together with their frequencies, and at deriving association rules like, for example, having experienced the subsequence first job, first union, first child, is most likely to be followed by a sequence marriage, second child. By contrast, indeed, mining frequent sequences and rules also reveals atypical life courses. Note that event sequences differ from state sequences as considered by Markov models or optimal matching. Nevertheless, sequence mining could as well be applied to state sequences. Technically, the mining of frequent event sequences and sequential association rules is a special case of the mining of frequent itemsets and association rules. In data mining, an association rule is just a rule that says that if A occurs then B is very likely to occur too. The basic tuning parameters of the mining process are the support and the confidence thresholds. The support is the minimal frequency in the data base for an itemset to be selected, while the confidence of the rule is the probability that the consequence occurs when the premise is observed. These basic selection criteria are complemented by other additional interestingness measures, like for example the proportion of the rule counter examples. Most algorithms for seeking frequent itemsets and rules are variants of the well known Apriori algorithm (Agrawal and Srikant, 1994; Mannilia et al., 1994). A typical application consists in finding the items that are more often ordered together by customers. Sequences that we consider here differ from general itemsets in that order matters. Multiple algorithms adapted for sequences have been proposed since the pioneering contributions by Agrawal and Srikant (1995) and Mannila et al. (1997). Illustration. We have not yet ourself experienced a sequential rule mining analysis on historical demographic data. For the sake of illustration, we shortly report here the analysis carried out by Blockeel et al. (2001). The data considered originated from the 1995 Austrian Fertility and Family Survey (FFS). The events analysed are those of the partnership and fertility retrospective histories of 4,581 women and 1,539 men aged between 20 and 54 at the survey time. The observed women and men were partitioned into 5 years cohorts and the objective of the analysis was to discover frequent partnership and birth event patterns that mostly varied among cohorts.

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The mining was done by means of the Warmr process implemented in the ACE Data Mining System (Blockeel et al., 2004). The search was not limited to simple sequences of strictly ordered events but allowed for more complex patterns combining multiple subsequences. A important found pattern was for instance having a child after first union and having both a marriage and a second child after this first birth, the marriage and second child being not ordered. The seeking of such not strictly ordered pattern requires indeed some filtering, namely the elimination of redundant patterns. For example, completing the above mentioned pattern with the additional condition of having a marriage after the first union would not brought any new information and is therefore redundant. Also, the rules generated were restricted to premises refereing to the cohort. Finally, only patterns that exhibit a great discrepancy in the proportion of individuals satisfying it in each cohort were retained.

0.75 0.7 0.65 0.6

Frequency

0.55 0.5 0.45 0.4 0.35 0.3 0.25 40

42

44

46

48

50 Cohort

52

54

56

58

60

Fig. 2. Negative trend in the proportion of first unions starting at marriage (Figure reproduced from Blockeel et al., 2001, with permission of the authors)

Figure 2 is an example of outcome provided by this analysis. It shows the strong declining proportion of individuals that started their first union when they married. The mining process found this pattern, i.e. date of first union equals date of marriage, to be the one that exhibits the strongest changes in frequency among cohorts. Indeed, many other sometimes more complicated patterns were found to also have great variability in their frequency. 3.2

Social transition analysis with induction trees

Let us now turn to induction trees and the insight they may provide on the understanding of social mobility. In mobility analysis, the focus is on how states at previous time t − 1, t − 2, . . . and possibly some additional covariates influence

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15

the present state at t. This setting is very similar to that of Markovian models. In contrast with this parametric modeling approach, the tree induction is, however, a non parametric method. It provides a heuristical way to catch how the previous states and covariates jointly influence the state at t. Though we focus here on social mobility analysis, it is worth mentioning that the scope of induction trees for life course analysis is much broader. For instance, De Rose and Pallara (1997) used a tree approach for segmenting time to marriage curves of Italian women, Billari et al. (2000) used trees for analysing differences in event sequences between Austrians and Italians, and we can easily imagine many other applications. Induction trees, i.e. decision trees induced from data, are basically supervised classification tools (Quinlan, 1986). As pointed out in Ritschard and Zighed (2003), they convey also powerful descriptive information. Their learning principle is quite simple and they produce easily interpretable results. An induced tree defines rules for predicting the value of a response variable from a set of potential predictors. The set of rules characterizes indeed a partition of the cases, each rule defining a class. The prediction inside each class of this partition is simply the modal observed value when the response is categorical and the mean observed value when it is quantitative. In the quantitative case, the tree is called a regression tree (Breiman et al., 1984). Extension in this case include model trees (Malerba et al., 2002) and logistic model trees (Landwehr et al., 2003), which use a linear or logistic regression for the prediction inside the classes of the partition. Tree algorithms have also been proposed for predicting functions instead of values and those that like RECPAM (Ciampi et al., 1988) predict for instance survival functions may be of special interest for life course analysis. Here we consider only categorical responses, i.e. classification trees. The easiest way to describe the tree induction principle is by looking at an example. We begin therefore by describing the framework of the illustration we will consider. We use data on intergenerational social transition in the 19th century Geneva (Ryczkowska and Ritschard, 2004). The data were collected from the marriage registration acts that provide the profession of the spouses as well as that of their parents. For 572 acts, it has been possible to find a match with the marriage of the father of one of the spouses. For these cases, we have the profession of the married man, of his father at the son’s marriage, of the matched father at his own marriage and of the grand-father at the matched father marriage. The professions were grouped into three social statuses, namely low, high and clock and watch makers who formed a important specific corporation in the 19th century Geneva. The variable we want to predict is the status of the son at his marriage, which is clearly a categorical response, and we consider four potential predictors. The first three are status variables, namely the status of the father at son’s marriage, the status of the matched father at his own marriage and the status of the grand-father at father’s marriage. The fourth predictor is the birth place that can take one of 12 values: Geneva city (GEcity), Geneva surrounding land (GE-

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land), neighboring France (neighbF), Vaud (VD), Neuchatel (NE), other French speaking Switzerland (otherFrCH), German speaking Switzerland (GermanCH), Italian speaking Switzerland (TI), France (F), Germany (D), Italy (I) and other. The grown tree is shown in Figure 3. The tree growing principle is as follows. First, all cases are grouped together in a root node (at the top of the tree) in which the distribution of the response variable, the status of the married man for our analysis, is its marginal distribution. The goal is to split this group in new nodes such that the distribution of the response variable differs as much as possible from one node to the other. The splitting is done iteratively using the categorical values of the predictor selected at each step. At the first step, we seek the predictor that best splits the root node and split it according to its values. The process is then repeated at each new node until a stopping rule is reached. Stopping rules typically concern the minimal node size, the maximal number of levels or the statistical significance of the improvement in the optimized criterion. In our study, we have retained the CHAID method (Kass, 1980) that selects at each step the predictor that, when it is cross tabulated with the response variable, generates the most significant independence Chi-Square statistics. CHAID also seeks the aggregation level of the categories of the predictors that generates the most significant Chi-square and splits then indeed according to the optimally merged categories. We generated the tree of Figure 3 with Answer Tree 3.1 (SPSS, 2001) by setting the minimal node size to 15 and requiring a maximal significance level of 5%. Alternative methods, among which CART proposed by Breiman et al. (1984) and C4.5 due to Quinlan (1993) are among the best known, differ mainly by the criteria used for selecting the split variable at each step. CART maximizes the reduction in the Gini index also known as the quadratic entropy. It generates only successive binary splits. C4.5 uses the gain ratio defined as the reduction in Shannon’s entropy normalized by the entropy of the distribution among the classes of the generated partition. Unlike the CHAID method, for which the significance of the Chi-square provides a natural validation for the split, CART and C4.5 do not have such a natural split validation criteria. These methods complete therefore the growing process with a post pruning round that, starting from the leaves, eliminates unreliable splits. Only splits that improve the predictive error rate are retained. There are also graph induction tools, among which SIPINA (Zighed and Rakotomalala, 1996), that generalize trees by allowing the merge of nodes with similar inside distribution. Knowledge provided by the tree. Looking at Figure 3, we see that the first split is done according to the father status at son’s marriage. This tells us that among the four attributes considered, the status of the father is the most discriminating. The status of the married man depends for instance more on the father status than on his birth place. The distribution inside the nodes of the first level are just the columns of the cross classification of the statuses of the father and the son. We observe here that the clock makers form a very closed group with a high probability for the son to become a clock maker when the father

high

high;deceased Node 19 Category % n low 10.00 2 clock 25.00 5 high 65.00 13 Total (3.50) 20

clock;low

gd-father (father's mar) social status (3 cat) P-value=0.0330, Chi-square=6.8202, df=2

clock;low Node 13 Category % low 16.00 clock 30.00 high 54.00 Total (8.74) 8 15 27 50

n

Node 6 Category % n low 66.67 10 clock 0.00 0 high 33.33 5 Total (2.62) 15

Father (his mar) social status (3 cat) P-value=0.0050, Chi-square=10.5961, df=2

neighbF;GermanCH;VD

Node 5 Category % n low 11.21 13 clock 18.97 22 high 69.83 81 Total (20.28) 116

birth place (11) P-value=0.0000, Chi-square=28.8085, df=2

GEland;GEcity;F;other;D;othFrCH;I

Node 12 Category % n low 7.58 5 clock 10.61 7 high 81.82 54 Total (11.54) 66

Node 18 Category % n low 6.52 3 clock 4.35 2 high 89.13 41 Total (8.04) 46

high Node 1 Category % n low 17.56 23 clock 16.79 22 high 65.65 86 Total (22.90) 131

clock

Node 7 Category % low 10.00 clock 24.00 high 66.00 Total (8.74)

high

deceased Node 3 Category % n low 19.46 50 clock 35.02 90 high 45.53 117 Total (44.93) 257

clock;low

GEland;F;TI;VD;D Node 15 Category % low 16.18 clock 57.35 high 26.47 Total (11.89)

n 11 39 18 68

GEcity;neighbF;GermanCH;NE;I

birth place (11) P-value=0.0180, Chi-square=8.0353, df=2

n 18 44 28 90

deceased

Node 16 Category % low 17.53 clock 29.90 high 52.58 Total (16.96)

n 17 29 51 97

GEland;GEcity;neighbF;GermanCH;I

high Node 20 Category % n low 14.29 5 clock 14.29 5 high 71.43 25 Total (6.12) 35

clock;low Node 21 Category % low 19.35 clock 38.71 high 41.94 Total (10.84)

n 12 24 26 62

Node 17 Category % n low 50.00 10 clock 25.00 5 high 25.00 5 Total (3.50) 20

F;other;VD;D

birth place (11) P-value=0.0057, Chi-square=10.3488, df=2

Node 9 Category % n low 23.08 27 clock 29.06 34 high 47.86 56 Total (20.45) 117

Father (his mar) social status (3 cat) P-value=0.0143, Chi-square=8.4928, df=2

gd-father (father's mar) social status (3 cat) P-value=0.0005, Chi-square=19.8163, df=4

Node 8 Category % low 20.00 clock 48.89 high 31.11 Total (15.73)

Node 14 Category % n low 31.82 7 clock 22.73 5 high 45.45 10 Total (3.85) 22

n 10 53 20 83

n 5 12 33 50

Node 2 Category % low 12.05 clock 63.86 high 24.10 Total (14.51)

father (son's mar) social status (3 cat) P-value=0.0000, Chi-square=77.4656, df=6

Node 0 Category % n low 21.33 122 clock 34.27 196 high 44.41 254 Total (100.00) 572

son (his mar) social status (3 cat)

low

Node 10 Category % low 32.10 clock 35.80 high 32.10 Total (14.16)

GEland;GEcity;F

n 26 29 26 81

Node 11 Category % n low 65.00 13 clock 10.00 2 high 25.00 5 Total (3.50) 20

neighbF;GermanCH;VD;othFrCH

birth place (11) P-value=0.0163, Chi-square=8.2390, df=2

Node 4 Category % n low 38.61 39 clock 30.69 31 high 30.69 31 Total (17.66) 101

Life Course Data in Demography

Fig. 3. Social transition tree with birth place covariate

17

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G. Ritschard and M. Oris

is himself clock maker while this probability is much lower for the three other groups. A similar results holds for the high classes, while there are evidences about social ascension possibilities when the father belongs to the lower class. Three of the four first level nodes are split further. The only one that is not split is that of married men whose father belongs to the clock and watch makers. This node is thus a terminal leaf, which indicates that the status of clock makers fathers conveys all the significant information for predicting the status of the son. This is a consequence of the strong social reproduction process inside the class of clock makers. The married men whose father was deceased are split according to the grand-father’s status, which means that the grandfather status is more discriminating for this subgroup than the status of the father at his own marriage. There is a strong tendency for the married man to reproduce the grand-father’s status when the father is deceased. The group defined by a high status of the father as well as that defined by a low father’s status are split according to the birth place. Both splits are binary. They do not make use, however, of the same binary partition of birth places. In both cases, i.e. with a father belonging to the low or high classes, the men born either in neighboring France, in German speaking Switzerland or in Vaud have a relatively high probability to get only a low status. This is also true for men born in French speaking Switzerland outside Geneva and Neuchatel when their father belongs to the lower class. The additional levels show that when the high position of the father results from a recent social ascension, i.e. an ascension since the father marriage (level 3) or from the position of the grand-father (level 4), the reproduction of the father status by the married man is less strong. The subtree that concerns the men whose father deceased, shows effects of the grand-father status very similar to those of the status of the father when he is alive at the marriage. Goodness-of-fit of the descriptive tree. Classically, the quality of a tree is evaluated in terms of its classification predictive quality, which is measured by the tree correct classification rate. Recall that the classification is done by assigning to each case the most frequent value in its leaf. For our tree, the correct classification rate is 57.6%. This corresponds to a 42.4% error rate. At the root node, before introducing any predictor, the correct classification rate is 44.4%, which gives an error rate of 55.6%. Our tree reduces thus the error rate by 24%. These figures are, nevertheless, irrelevant in our case, since we are not using the tree for classification purposes. We do not consider the classification results. The descriptive knowledge considered follows directly from the distributions inside the nodes. Hence, we consider the tree as a probability tree rather than a classification tree. In Ritschard and Zighed (2003) we have proposed indicators that better suit this descriptive point of view. We can for instance measure with a Likelihood-Ratio Chi-square (G2 ) the divergence between the distributions predicted by the tree (those in the leaves) and those of the finest partition that our four predictors may generate. We get 312.5 for 300 degrees of freedom, and its p-value is 29.8% indicating clearly a good fit. Note that though the four predic-

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tors define theoretically 576 different profiles, only the 163 actually observed are taken into account. Table 6. Goodness-of-fit of the tree and subtrees Tree Indep Level 1 Level 2 Level 3 Fitted Saturated

G2 482.3 408.2 356.0 327.6 312.5 0

df 324 318 310 304 300 0

sig 0.000 0.000 0.037 0.168 0.298 1

BIC 2319.6 1493.9 1492.5 1502.2 1512.5 3104.7

AIC 812.3 750.2 714.0 697.6 690.5 978.0

pseudo R2 0 0.14 0.23 0.28 0.30 1

For comparison purposes, Table 6 reports the G2 Chi-square statistic for a set of nested trees, namely the independence tree corresponding to the root node only, the tree expanded respectively one level only, two levels and three levels, the fitted tree and the saturated tree that generates the finest partition. Beside the G2 , its degrees of freedom and significance level, the table shows the BIC and AIC information criteria and the adjusted pseudo R2 . The latter measures the percentage of reduction of the G2 /df ratio as compared with the independence tree. The BIC (= G2 + ln(n)p, with p the number of parameters and n the number of cases) and the AIC (= G2 + 2p) are G2 ’s penalized for the complexity. We see that with less than three levels there is a lack of fit, the divergence with the finest partition being significant at the 5% level. The difference in G2 ’s between two nested trees can also be compared with a Chi-square distribution with degrees of freedom being the difference in these degrees for the two models. Thus, the Level 3 tree differs by ∆G2 = 15.1 and ∆df = 4 from the fitted model, which is clearly significant. Hence, the two splits leading to level 4 look jointly statistically significant. From the BIC point of view, the level 2 tree provides the best compromise between fit and complexity. Level 3 or 4 trees seem however preferable according to the interesting insight brought by the additional levels and the significant divergence of level 2 with the saturated model. The AIC, which is known however to underestimate the impact of complexity, selects here the fitted tree.

4

Conclusion

This paper stressed the scope and limits of various methods available for analysing life course data. The discussion is by no means exhaustive. Our aim was to illustrate different approaches, and especially the emerging data mining techniques that should be able to provide original additional insights on results provided by more classical statistical methods. Among the techniques we did not discuss, optimal matching (Abbot and Forrest, 1986; Malo and Munoz, 2003) deserves

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special attention. Optimal matching is, like Markovian models, a state sequence analysis tool. It is merely a data mining approach, since it proceeds heuristically. Unlike the mining of frequent sequences that does not care about the similarity between sequences, optimal matching is concerned with the discovering of similarities between sequence patterns. Optimal matching evaluates the proximity between two sequences by seeking the minimal number of changes that can transform a sequence a into a sequence b. Survival (Ciampi et al., 1988; Segal, 1988) and risk trees (Leblanc and Crowley, 1992) developed in the field of biomedicine during the first half of the 90’s would also merit further attention from historians and demographers. It is worth mentioning that the statistical and data mining approaches are not substitutes for one another. They are complementary, each method bringing its own insight. The choice of a method will be dictated by the kind of data available: spell durations, event sequences, state sequences, and indeed the type of results expected: knowledge about probability of transitions, effects on these risks, characteristic trajectories or life sequences. An other important element for this choice, at least for the end user, is the availability of user-friendly softwares and the level of expertise required to run the method and interpret the results. Many softwares propose duration or hazard models and/or classification trees. It is less obvious to find friendly tools for Markov models and the mining of sequential rules. March 2 is a promising solution for Markov models, while specialized softwares like Clementine propose sequence mining tools (see http://www.kdnuggets.com for a list of data mining softwares). The use and interpretation of hazard models is very similar to that of other regression like models, which renders them attractive. The interpretation of induction trees is also very straightforward and looks therefore as a promising tool. Nevertheless, the fine tuning of trees, which may be highly instable, requires generally more care than hazard models. Mining frequent sequential patterns requires also some experience to get interesting patterns. In any case, the new highlights provided by these data mining approaches are worth the effort.

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